1. Field of the Invention
This invention relates to rotary machines utilizing planetary motion to either pump fluid or be driven by fluid or accomplish both simultaneously.
2. Description of the Related Art
The fundamental starting point for this invention is the motion of the Wankel type engine. Technically such an engine is a planetary motion machine, which one inventor characterized as: xe2x80x9ca rotating piston arrangement where a motor is guided by a gear mechanism meshing with a toothed reaction wheel in such a way that the rotor can move into or out of one or more consecutively following work chambers which accommodate rotor and are in a stationary casing.xe2x80x9d F. Jernaes, U.S. Pat. No. 3,221,664, Dec. 7, 1965.
A planetary motion machine offers the benefit of fewer moving parts than a typical machine using cyclical motion, valves, or conversion from rotary to linear motion or vice versa to exert or receive pressure. A planetary motion machine may be a pump (that is taking in a fluid stream and compressing it to be exhausted at higher pressure), or a turbine (utilizing pressure to drive a rotor circularly to a lower pressure exhaust, and generating rotary mechanical power in a rotating shaft). A planetary motion machine has less eccentric motion than a typical straight piston machine. It has fewer moving parts in part because the machine is inherently a rotary machine and need not convert linear motion to rotary motion. Its disadvantages are that traditionally the classic planetary motion machine has only one compression per rotor cycle, and at high speed, there can be problems maintaining a seal of the compression chambers.
A classic planetary motion machine is illustrated in FIG. 2. The basic shape of the chamber, looking at the chamber from the xe2x80x9ctopxe2x80x9d parallel to the axis of the rotating parts, is that of a symmetric peanut, though the xe2x80x9cwaistxe2x80x9d of the peanut is barely narrowed. The peanut shape is called a peritrochoid in mathematics. The rotor looks like an equilateral triangle with symmetric bulged sides. In essence, the rotor, to use a layperson""s description, rolls around in the inside of the peanut with each apex in contact with the peanut. If an engine is placed on the drive shaft of the planetary machine, it will cause the rotor to spin, and the action of an alternating increase and decrease in volumes of the working chambers in combination with alternate occlusion and exposure to intake and exhaust ports will cause fluid to be pumped. Alternatively, if pressurized fluid is allowed into a chamber to force the rotor to turn, then the drive shaft will be forced to rotate and will produce mechanical power at the shaft. Similarly, if pressurized fluid is allowed into a chamber to force the rotor to turn, by changing the position of the intake and exhaust ports for a different chamber, that different chamber can be used to compress fluid, effectively permitting the rotary machine to be a compressor and turbine simultaneously. The fluid can be liquid or gas or a combination.
In order to make a planetary machine attractive, scientists have sought to have more than one chamber simultaneously performing compression/exhaustion while another chamber performs induction/expansion during each rotation of the rotor, and at the same time minimize the number of moving parts, and minimize the speed of what parts are moving. The machine in the present invention is a double pumping or double action planetary machine, meaning that for each planetary cycle, the machine can have one chamber perform a function of compression/exhaust or intake/expansion, while another chamber performs another function of either compression/exhaust or intake/expansion, and therefore the cycle of at least one chamber consists of a) two motions of intake/compression/exhaust, b) two motions of intake/expansion/exhaust or c) one action of each of intake/compression/exhaust and intake/expansion/exhaust.
In 1976, Whitestone, U.S. Pat. No. 3,998,054, Dec. 21, 1976, was issued a patent for a xe2x80x9cRotary Mechanism with Improved Volume Displacement Characteristics.xe2x80x9d While claiming improved displacement characteristics, and using ports in side plates, his rotor did not use the device of a duct through the rotor face and thence to a side port, nor did his pump contemplate a two-lobe peritrochoidal cavity. The effect of not using this duct or aperture through the rotor face and the lack of two-lobe peritrochoidal cavity is that for any given planetary cycle, the pump fails to achieve the swept volume and compression ratio (maximum volume to minimum volume) that the present invention achieves. This can be seen by reviewing FIGS. 1 through 8 in Whitestone ""054. The advantage of the present invention is that a working chamber is nearly totally evacuated from a maximum volume. In Whitestone, particularly as the geometry of his proposed rotor veered away from the three lobed rotor in a square cavity in FIG. 2, Whitestone""s invention faces one of two efficiency difficulties. First, there is a large permanently retained minimum volume 25f as in FIG. 9E, which minimizes the compression ratio of the maximum to minimum volume. Alternatively, second, there is a relatively small maximum volume with a somewhat smaller but substantial minimum volume 12af as in subfigures CF and DF of FIG. 13, but no port available for exhaust in Whitestone""s ""054 invention. Whitestone""s porting, shown in Whitestone ""054 FIG. 9a, which is the identical rotor position to Whitestone ""054 subfigure CF of FIG. 13, particularly for a solid rotor which eliminates volume 25f of FIG. 9E, shows the traditional geometric difficulty faced by Maillard, United Kingdom (British) Pat. No. 583,035 issued Jan. 2, 1947, and prior art rotary pumps of either a) maximizing intake volume for the beginning of compression, but also enlarging the volume being compressed at time of exhaust, as in Whitestone ""054, or b) lessening intake volume for the beginning of volume, and lessening the volume being compressed at time of exhaust. An example of the latter is Maillard UK Pat. 583,035 and Juge, U.S. Pat. No. 3,869,863, Mar. 11, 1975.
A rotary pump was proposed in an unpublished project proposal at the University of Calgary, Alberta, Canada referred to as a Zwiauer-Wankel configuration of rotary Stirling engine, for which a figure is shown at p. 79 of G. Walker, Stirling Engines, Clarendon Press, Oxford 1980, Library of Congress Call No. TJ765.W35, and is described at p. 115 of that book, Walker, Stirling Engines. In G. Walker, et al, The Stirling Alternative: Power Systems, Refrigerants and Heat Pumps, p.78, (Gordon and Breach Science Publishers 1994), the same author remarks that the Zwaiuer-Stirling rotary engine is an xe2x80x9carrangement [that] could provide a compact high specific output machine but although proposed over 20 years ago it has not been reduced to practice so far as is known.xe2x80x9d From the drawing, Zwaiuer appeared to use a solid rotor form with porting after the fashion of Maillard or Whitestone ""054, and in any event did not contemplate the use of a duct through the rotor and corresponding porting arrangement.
There are several other planetary machines which do not achieve double action where ducts through the rotor are contemplated, and/or where the maximum to minimum volume (the compression ratio) is not particularly useful for efficient fluid flow, and/or there are sealing problems. However, no art utilizes a system set out in this invention involving ducting, porting and the relative position of the rotor, duct and ports for the basic pumping or turbine action of the planetary machine to achieve double action with a superior volumetric efficiency without seal loss, double action in a three vaned-two lobed pump meaning two compressions and two expansions of fluid per planetary cycle. Maillard, United Kingdom (British) Pat. No. 583,035 issued Jan. 2, 1947, recognized the geometric constraints of his design, but absent a fluid passage through the rotor and proper design of ports and proper location of such a fluid passage, he could not overcome the geometric constraints. The present invention successfully hurdles the geometric constraints and achieves double action which none of the prior art has achieved, and with a minimization of moving parts. See, for example, ducts through the rotor, but no double action: Child, U.S. Pat. No. 4,986,739, Jan. 22, 1991, White, Jr., U.S. Pat. No. 4,872,819, Oct. 10, 1989, Nakayama, U.S. Pat. No. 4,345,886; ducts for lubrication or cooling: Miles, U.S. Pat. No. 4,097,205, Jun. 27, 1978, Nakayama, U.S. Pat. No. 4,345,886, Aug. 24, 1982 (using retractable vanes in the housing).
One of the best uses of this particular invention is as a pump powering an aircraft gyroscope. Non-electric gyroscopes are powered by an air stream that expands through a small turbine that drives the gyroscope. Failure of the pump interrupts the flow of air and causes the gyro to slow down and tumble. Slow or tumbling gyros will deliver incorrect navigational information. A typical pump using sliding vanes made of carbon graphite is seen in Kaatz, U.S. Pat. No. 3,191,852, Jun. 29, 1965, and Bishop, U.S. Pat. No. 5,181,844, Jan. 26, 1993, U.S. Pat. No. 4,820,140.
Apex seals may be kept in close contact with a roughly orthogonal surface using centrifugal force as seen in Kaatz, U.S. Pat. No. 3,191,852, Jun. 29, 1965, and Bishop, U.S. Pat. No. 5,181,844, Jan. 26, 1993, U.S. Pat. No. 4,820,140, or using a technique of feeding pressured air in behind the vanes as seen in Smart et al, U.S. Pat. No. 4,804,313, Feb. 14, 1989. Springs can also be used.
Optimum self-lubricating materials can be seen in any number of patents using polytetraflouroethylene (PTFE), or better yet using carbon fiber reinforced polyetheretherketone (PEEK), particularly continuous carbon fiber reinforced PEEK. Other materials usable as self-lubricating materials are set out in Davies et al, U.S. Pat. No. 5,750,620, May 12, 1998.
The term continuous carbon fiber reinforced PEEK is focused on polyetheretherketone, and a close material cousin PEKK, polyetherketoneketone, but the term includes a compound selected from the group of polyaromatic compounds having amorphous crystal structure corresponding in intermolecular distance to the intermolecular distance of continuous carbon graphite crystal structure such that upon melting of said polyaromatic compound having amorphous crystal structure in the presence of continuous fiber carbon graphite, said combination results in carbon crystal lattice reinforcement of said polyaromatic compound.
This invention has three major features yielding improved performance.
First, the creation of a duct through the rotor to the curved facexe2x80x94that is, diagonally from the side of the rotor through the rotor to the curved face of the rotorxe2x80x94yields, in combination with carefully arranged ports, double pumping action and enhanced inlet or exhaust porting. By proper arrangement of the location of the inlet parts, duct and exhaust parts, no new moving parts are introduced beyond the classic rotary machine design, yet a double action pump is created with substantially improved compression ratio. If pressured air is delivered to the invention with a differential lower pressure on the xe2x80x9coppositexe2x80x9d side of the pump, a double pumping turbine yielding power to a drive shaft results with a favorable compression ratio.
Second, the planetary machine""s performance is enhanced by the use of modem self-lubricating plastics to achieve better sealing.
Third, the use of a volute, in conjunction with the port(s). A volute is a spiraled air pipe that improves the intake and outflow characteristics when collecting and delivering air to the working volumes of the planetary pump. The volute reduces losses caused by turbulence at sharp corners, elbows, etc. and losses caused by sudden expansion.
The invention achieves a variety of objectives by this design. The invention can be a pump when an engine or other rotating device is connected to the machine and causes the rotor to rotate, forcing fluid through the parts of the machine. One preferred use is a vacuum air pump with the drive shaft driven by an airplane engine causing the rotor to turn, which draws air through a gyroscope.
The invention may be a turbine when pressurized fluid drives the machine, or an engine when combustible mixture is ignited in the working chambers. The invention will be described in terms of a pump, understanding the claims are no limited to a pump and that if, a pressure differential between the intake and exhaust side of the pump exists, the machine will function as a turbine.
The general characteristic of the preferred embodiment is somewhat like a two rotor NSUxe2x80x94Wankel internal combustion engine found in some automobiles and aircraft except that the preferred embodiment of the invention disclosed herein is in the form of a mechanically driven pump which delivers air to or from air-driven gyroscopic attitude instruments for piston engine powered aircraft. The pump is composed of a forward stationary side plate with mounting fixture, a two-lobe peanut-shaped peritrochoid stator shroud within which rotates a three-face triangular-shaped rotor, a port plate with intake and exhaust orifices, a volute for ducting the air to and from the external pump connections and the working volumes, a second port plate, stator and rotor combination, followed by a rear stationary side plate. The entire unit is held together by four symmetrically placed bolts.
The center of each cam is displaced eccentrically from the center of the driving shaft. Each cam rotates within a hole machined into the center of each rotor and drives, in the preferred embodiment, continuous carbon fiber reinforced PEEK bearings fit into each rotor, which action in turn causes rotor rotation as later described.
While reference is made to PEEK in the preferred embodiment, PEKK (polyetherketoneketone) has similar properties. More broadly, the invention preferably utilizes for either the bearings and/or or the rotor apex tips as described a compound selected from the group of polyaromatic compounds having amorphous crystal structure corresponding in intermolecular distance to the intermolecular distance of continuous carbon graphite crystal structure such that upon melting of said polyaromatic compound having amorphous crystal structure in the presence of continuous fiber carbon graphite, said combination results in carbon crystal lattice reinforcement of said polyaromatic compound. Enhancements to strength and lubrosity occur upon curing, including curing under pressure. Such compound, including PEEK and PEKK, achieving carbon crystal lattice reinforcement in such manner will be referred to a continuous carbon fiber reinforced polyaromatic compound. Even more broadly, the continuous carbon fiber reinforced polyaromatic compounds, as defined, and those elastomer reinforced polymeric compositions referenced in Davies, U.S. Pat. No. 5,750,620, will be referred to collectively as carbon fiber reinforced polymeric compositions. Materials such as scintered bronze impregnated with PTFE along with carbon fiber reinforced polymeric compositions, or even hydrocarbons in certain applications, will be the broadest category of suitable materials and will collectively be called self-lubricating materials. All of these may be used, but the optimum selection for use is a continuous carbon fiber reinforced polyaromatic compound such as continuous carbon fiber reinforced PEEK.
The preferred embodiment of each rotor has three apices, and therefore three faces corresponding to the number of apices. Each set of two adjacent apices and the intervening face can be referred to generically as a lobe and will have a working chamber of varying volume opposite that lobe which will be moving rotationally and varying volume simultaneously. The rotor is composed of hardened aluminum, e.g., 6061-T6 and machined to the desired contour of three triangularly placed arcs. Each of the three faces of said rotor is penetrated by one of the important innovations of the claimed invention: namely a single duct machined or molded through the rotor face which pierces the side of the rotor which is orthogonal to the face. The duct then forms an aperture through which air flows undisturbed when both ends are not obstructed. The rotor also contains an annular timing gear affixed to either side. This annular gear meshes with a stationary sun or spur gear fixed to the non-rotating forward and rear side plates of the pump and constrains the rotor motion to the desired planetary cycle, much like the Wankel design (The gears could be replaced by a guide similar to Grey""s invention U.S. Pat. No. 3,884,600, May 20, 1975.
In the preferred mode, each apex of the rotor is machined with a groove to accept an apex seal. The apex seal is a rectangular strip composed of a self-lubricating continuous carbon fiber reinforced PEEK material. The apex seal can be pressed against the shroud by way of compression springs. The spring constant and the amount of compression are chosen such that the mechanical properties of the PEEK apex seals are not exceeded. The apex seal forms a zero clearance sliding contact point with the stationary peritrochoid shroud which guarantees that each working volume defined by each rotor face operates independently with minimal exchange of air. The apex seal can be pressed alternatively or additionally against the shroud by means of compressed air fed in behind the apex seal in the manner suggested by Smart, U.S. Pat. No. 4,616,985, Oct. 14, 1986, again such a way that the mechanical properties of the PEEK apex seals are not exceeded. Smart proposes that air be fed in behind the sliding vanes in his pump for purpose of equalizing pressure.
The peritrochoid shrouds are made of hardened aluminum like 6061-T6, preferably with hard-coat anodizing, and with the next-described side plates form the cavity within which each rotor rotates. The peritrochoid shroud and rotor lie between two side plates, either of which may be ported, but for purposes of the best mode, one of which is a port plate and the other an unported side plate. There can be two port plates as an alternative. The side plates are disposed in conjunction with the shroud such that the side plates are in sliding contact with the rotor. The side plates on which are disposed the stationary sun gears are also made of aluminum and mate with the peritrochoidal shrouds. The side plates could be made of or coated with a self-lubricating material such as PEEK, particularly where there is relatively high speed relative motion between the side plates and the rotor. The side housing could be of PEEK, but this is a less desirable equivalent than the vanes being made of PEEK which are much smaller, and the side housing not being made of PEEK. The sun gears, peritrochoidal shrouds, annular gears and rotors are specifically oriented such the planetary motion of the rotor apices is exactly contained by the shroud. To maintain low friction, the side plates, including the port plate(s) can be made from continuous carbon fiber reinforced PEEK similar to the apex seal material. In this way, all sliding surface contacts use low friction self-lubricating material.
Opposite and parallel to the side plates are the port plates which contain two intake ports symmetrically placed about the central axis coincident with the driving shaft and the shroud longitudinal center line and two exhaust ports also symmetrically placed about the central axis. The intake and exhaust ports are of sufficient cross sectional area that the air flow will not choke (reach Mach 1) during normal operation which would reduce performance. The position of the ports is determined to maximize the flow rate performance but generally, in a pump where the fluid will be exhausted from a working chamber and out through a duct in the rotor face to an exhaust port in the side plate, the intake port on a side plate is positioned and configured in such a way that:
a) the intake port is covered by the rotor side at all times except between the xe2x80x9cintake port openxe2x80x9d and the xe2x80x9cintake port closedxe2x80x9d rotor position at which time there exists an unobstructed path for air to flow from the intake volute to the working volume formed by the shroud, the side plates, and the rotor face exposed to the intake port. The ports in this configuration are located inside the outer bound of the rotor, but outside the innermost trace of the face of rotor during the rotation cycle.
b) the xe2x80x9cintake port openxe2x80x9d rotor position is that rotor position where the working volume is near its minimum and the exhaust port is closed or occluded.
c) the xe2x80x9cintake port closedxe2x80x9d rotor position is that rotor position where the working volume is near its maximum and the exhaust port is closed or occluded.
The exhaust port is positioned and configured in such a way that:
a) the exhaust port is covered by the rotor side at all times.
b) between the xe2x80x9cexhaust port openxe2x80x9d and xe2x80x9cexhaust port closedxe2x80x9d rotor position, the exhaust port is aligned with the rotor side aperture formed by the claimed invention of a duct piercing the rotor face previously exposed to the intake port. The alignment is such that an unobstructed path is formed for air to flow from the working volume to the exhaust volute and subsequently out of the device entirely.
c) the xe2x80x9cexhaust port openxe2x80x9d rotor position is a position after the working volume is near its maximum and the intake port is closed, and some contraction of the working volume has occurred so that the desired pressure is created.
d) the xe2x80x9cexhaust port closedxe2x80x9d rotor position is that where the working volume is near its minimum.
With respect to the mounting of a volute, each port plate covers one side of the volute. Each side of the volute contains two scroll-like channels which direct air to the two intake ports and from the two exhaust ports. Each volute channel provides an unobstructed, smooth conduit from the ports to the external connections of the pump. The volute is machined from aluminum and also contains a centrally located longitudinal hole through which the driving shaft rotates. The driving shaft can be supported here too by means of a self-lubricating bearing fit into the volute piece.
The actual operation of the pump as a fluid movement device begins with the main driving shaft rotating the rotor and a particular rotor face towards the xe2x80x9cintake port openxe2x80x9d position. The intake port is uncovered by the rotor side exposing the minimum working volume and a trailing rotor face to the intake volute. The rotor rotation produces an expanding volume which in turn produces a lower-than-inlet pressure which pulls air into the working volume through the intake volute. Air ceases to flow into working volume as the intake port is occluded by the rotor side prior to the working chamber volume contraction due to rotor rotation. As rotor rotation continues, the air is compressed in a now fully enclosed working chamber until the xe2x80x9cexhaust port openxe2x80x9d position when a clear path forms from the working volume to the exhaust port via the duct from the rotor face to the rotor side. Air continues to flow out of the contracting volume through the duct and into the exhaust volute until the xe2x80x9cexhaust port closedxe2x80x9d position is reached. This sequence also occurs in the second lobe of the peritrochoid shroud, albeit out of phase. Since the apex seals and side plates produce nearly zero clearance or actually zero clearance, there is little flow communication between the two lobes. Thus, with the claimed invention of an aperture or duct through the rotor face, the intake and exhaust ports can be utilized or be occluded based on maximizing volumetric efficiency rather than observing the geometric constraints found in the Maillard, United Kingdom Pat No. 583,035, Jan. 2, 1947 and Schwab, U.S. Pat. No. 4,551,073, Nov. 5, 1985 designs.
The intake ports, instead of being in the side plates, could be in the shroud, but the volumetric efficiency of the machine is significantly less.
For a turbine, there is no need to wait to create access to the exhaust port until after a period of contraction of a particular working chamber. The turbine can accept fluid to an expanding chamber immediately after minimal volume is achieved, cease accepting fluid to that chamber at maximum volume or in desired quantity, and have the chamber commence access to an exhaust port after an intake port is occluded, and after maximum volume has been achieved. Exhaustion of a chamber can continue until just before an apical tip is at a position where minimal volume is achieved.
The system can be a two rotor system which is statically balanced, and/or counterweights or cams may be added for dynamic balance. These counterweights can be fixed to the driving shaft beyond the forward and rear side plates. Multiple rotor combinations can be used to avoid large counterweights.
If the invention is to have each lobe have a separate exhaust stream, then each lobe must have its own separate exhaust duct and port; the above description of porting locations applies for each chamber, but to separate the exhaust streams, there must be more planning of the relative location of the exhaust ducts. Each duct must intersect the rotor side on a separate peritrochoidal track so that a particular duct only vents to a particular track. If a volute is desired, a volute for each duct and its corresponding track must be created.
There is no requirement in the invention that the duct through the rotor face be used for exhaust. The construct of the planetary machine may be inverted. The intake ports may be designed to be covered by the rotor side at all times, and located to be alternately exposed to an intake duct from the rotor side to the rotor face to a working chamber, with the exhaust ports alternately exposed to the working chamber when the intake ports are not exposed to the duct to the working chamber.
As a turbine, the invention has superior wear properties as a result of the continuous carbon fiber reinforced PEEK used.
Intake and exhaust ducts may be used carry fluid to or from intake and exhaust ports, rather than having ports opening during parts of the cycle directly to a working chamber. In this mode of invention, all ports will then be located inside the innermost trace in each chamber of the face of the rotating rotor.
If the lobes have their own separate exhaust duct and ports from each other, as suggested in the prior paragraph, the exhaust streams are separated, and if in the same way as the exhaust streams were separated the intake streams are separated, then the rotary machine can be set up by appropriate porting to be a pump and turbine, meaning one working chamber is pumping (intake from lower pressure and exhaust at higher pressure), while another is acting as a turbine (intake from higher pressure and exhaust at lower pressure). In essence, the pumping side will have an early close of intake in the rotor face motion for the working chamber acting as a pump and later opening and closing of exhaust, while the turbine side will have a relatively later close of intake in the rotor face motion for the working chamber acting as a turbine and later opening and closing of exhaust. More likely, the separation of the exhaust streams are separated, and, the intake streams are separated, there can be independent inputs and outputs for each respective working volume for specialized applications.
Alternatively, the intake and exhaust ports for one chamber can each have their own fluid source and exhaust outlet, and the intake and exhaust ports for an opposite chamber can each have their own fluid source and exhaust outlet. In that instance, one xe2x80x9csidexe2x80x9d or chamber can be acting as a compressor, with the other side acting as a turbine using the same previously-described principles for locating ports to achieve these effects.
Description of the Rotor Shape
The equations which describe the shape of the peritrochoid and the faces of the rotor are well developed in the open literature, Kenichi Yamamoto, Rotary Engine, Sankaido Co. Ltd. (1st ed. 1981), therefore only the results as they pertain to this embodiment are presented. The shape of the peritrochoid can be represented in orthogonal coordinates x and y by:
x=e cos a+R cos(a/3) 
y=e sin a+R sin(a/3) 
where a is the position angle of the main driving shaft and generates periodic motion every 1080 degrees of driving shaft rotation, e is the eccentricity, meaning the amount the rotor axis is displaced from the driving axis, and R is the radius of the rotor, meaning the distance from the rotor axis to the rotor apex.
The outer bounds of the shape of each rotor face in the preferred embodiment of a three lobe rotor can be represented by:       x    =                  R        ⁢                  xe2x80x83                ⁢        sin        ⁢                  xe2x80x83                ⁢        2        ⁢                  xe2x80x83                ⁢        θ            +                                    3            ⁢                          ⅇ              2                                R                ⁢        sin        ⁢                  xe2x80x83                ⁢        6        ⁢        θcos        ⁢                  xe2x80x83                ⁢        2        ⁢        θ            -              D        ⁢                  xe2x80x83                ⁢        cos        ⁢                  xe2x80x83                ⁢        3        ⁢        θsin        ⁢                  xe2x80x83                ⁢        2        ⁢                  xe2x80x83                ⁢        θ                  y    =                  R        ⁢                  xe2x80x83                ⁢        cos        ⁢                  xe2x80x83                ⁢        2        ⁢                  xe2x80x83                ⁢        θ            +                                    3            ⁢                          ⅇ              2                                R                ⁢        sin        ⁢                  xe2x80x83                ⁢        6        ⁢                  xe2x80x83                ⁢        θsin        ⁢                  xe2x80x83                ⁢        2        ⁢                  xe2x80x83                ⁢        θ            -              D        ⁢                  xe2x80x83                ⁢        cos        ⁢                  xe2x80x83                ⁢        3        ⁢        θcos        ⁢                  xe2x80x83                ⁢        2        ⁢        θ            
where the further variable D is found by the following equation as xcex8 varies from xe2x88x9230 to +30 degrees for each face and xcex8 is rotated in such range symmetrically about the rotor axis:   D  =      2    ⁢    e    ⁢                  1        -                              [                                          3                ⁢                e                ⁢                                  xe2x80x83                                ⁢                sin                ⁢                                  xe2x80x83                                ⁢                                  (                                      3                    ⁢                                          xe2x80x83                                        ⁢                    θ                                    )                                            R                        ]                    2                    
As the eccentricity e, in the limit, approaches zero, the three faces become closer to being arcs of a circle connecting the apices; however, the ideal compression ratio declines. The machine can also have three lobes.